^{1,a)}

### Abstract

This study considers the random placement of uniform sized spheres, which may overlap, in the presence of another set of randomly placed (hard) spheres, which do not overlap. The overlapping spheres do not intersect the hard spheres. It is shown that the specific surface area of the collection of overlapping spheres is affected by the hard spheres, such that there is a minimum in the specific surface area as a function of the relative size of the two sets of spheres. The occurrence of the minimum is explained in terms of the break-up of pore connectivity. The configuration can be considered to be a simple model of the structure of a porous composite material. In particular, the overlapping particles represent voids while the hard particles represent fillers. Example materials are pervious concrete, metallurgical coke, ice cream, and polymer composites. We also show how the material properties of such composites are affected by the void structure.

The author acknowledges valuable discussion with Dr. M. J. Buckley, CSIRO Mathematics, Informatics and Statistics.

I. INTRODUCTION

II. MICROSTRUCTURE FORMATION

III. RESULTS

IV. EFFECT ON MATERIAL PROPERTIES

V. CONCLUSION

### Key Topics

- Composite materials
- 26.0
- Crystal voids
- 21.0
- Materials properties
- 16.0
- Voids
- 10.0
- Elastic moduli
- 5.0

## Figures

Calculated specific surface area versus relative size of obstructing particles and voids, for different values of the porosity. Each point on the graph is the average of 50 simulations. In all of the cases shown here, the obstructing spheres occupy 20% of the volume.

Calculated specific surface area versus relative size of obstructing particles and voids, for different values of the porosity. Each point on the graph is the average of 50 simulations. In all of the cases shown here, the obstructing spheres occupy 20% of the volume.

Calculated specific surface area versus relative size of obstructing particles and voids, for different volume fractions of obstructing spheres (shown in the legend). Each point on the graph is the average of 50 simulations. In all of the cases shown here, the voids occupy 40% of the volume.

Calculated specific surface area versus relative size of obstructing particles and voids, for different volume fractions of obstructing spheres (shown in the legend). Each point on the graph is the average of 50 simulations. In all of the cases shown here, the voids occupy 40% of the volume.

Simulated configurations for different values of *R* _{o}/*R* being (a) 1, (b) 2, (c) 3. The black components are the obstructing spheres, the dark grey components are the overlapping spheres, and the light grey is the matrix phase.

Simulated configurations for different values of *R* _{o}/*R* being (a) 1, (b) 2, (c) 3. The black components are the obstructing spheres, the dark grey components are the overlapping spheres, and the light grey is the matrix phase.

Graphs showing the volume of isolated voids, relative to the volume of a single void sphere, for two different values of *R* _{o}/*R*. In each case there is 20% volume fraction of obstructing spheres and 30% volume fraction of voids.

Graphs showing the volume of isolated voids, relative to the volume of a single void sphere, for two different values of *R* _{o}/*R*. In each case there is 20% volume fraction of obstructing spheres and 30% volume fraction of voids.

Graph showing the number of isolated voids per unit volume, as a function of *R* _{o}/*R*, for different values of *ϕ* _{o}, as shown in the legend. The vertical dotted line marks the point *R* _{o}/*R* = 1.054, indicating the point at which obstructing spheres on a cubic lattice would preclude overlap of void spheres.

Graph showing the number of isolated voids per unit volume, as a function of *R* _{o}/*R*, for different values of *ϕ* _{o}, as shown in the legend. The vertical dotted line marks the point *R* _{o}/*R* = 1.054, indicating the point at which obstructing spheres on a cubic lattice would preclude overlap of void spheres.

Calculated effective permeability (left hand axis) and specific surface area (right hand axis) versus *R* _{o}/*R*, when *ϕ* _{o} = 0.2. Each point on the graph represents an average of 25 simulations.

Calculated effective permeability (left hand axis) and specific surface area (right hand axis) versus *R* _{o}/*R*, when *ϕ* _{o} = 0.2. Each point on the graph represents an average of 25 simulations.

Calculated effective Young's modulus (left hand axis) and specific surface area (right hand axis) versus *R* _{o}/*R*, when *ϕ* _{o} = 0.2. Each point on the graph represents an average of 25 simulations.

Calculated effective Young's modulus (left hand axis) and specific surface area (right hand axis) versus *R* _{o}/*R*, when *ϕ* _{o} = 0.2. Each point on the graph represents an average of 25 simulations.

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